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Polygons - Computer Graphics - Notes
The document discusses the definition and types of polygons, including convex, concave, and complex polygons. It outlines polygon representation, methods for inside testing, and algorithms for polygon filling such as boundary fill, flood fill, and scan line algorithms. Additionally, it compares the efficiency of different filling techniques and presents recursive methods for different fill algorithms.
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Introduces polygons, defining them as closed figures with sides and vertices. Lists types: Convex, Concave, Complex.
Defines convex polygons where lines between points stay inside, and concave polygons where lines may extend outside.
Describes complex polygons as those neither convex nor concave, including self-intersecting shapes.
Illustrates how polygons can be represented analytically with coordinates, showing specific points and their relationships.
Introduces methods to test whether a point is inside a polygon: Even-odd Method and Winding Number Method.
Explains the Even-Odd Method visually, showing how to determine if points are inside or outside a polygon.
Discusses the Winding Number Method, illustrating how to determine point containment using winding numbers.
Lists polygon filling algorithms including Boundary Fill, Flood Fill, Edge Fill, Fence Fill, and Scan Line.
Compares 4-connected and 8-connected methods for testing neighboring pixels in algorithms.
Presents the recursive Boundary Fill algorithm's procedure for filling areas inside polygons.
Details the Flood Fill algorithm, a method for filling adjacent pixels based on the current pixel color.
Describes the Scan Line Algorithm for filling polygons based on intersections of edges and managing pixel colors.
Compares key differences among Flood Fill, Boundary Fill, Edge Fill, Fence Fill, and Scan Line fill algorithms.
Details the Connected Boundary Fill Algorithm for faster pixel filling by considering all adjacent pixels.
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Polygons - Computer Graphics - Notes
- 1. Polygon (Definition) 1 Polygon isa figure having many slides. It may be represented as a number of line segments end to end to form a closed figure. The line segments which form the boundary of polygon are called as edges or slides of polygon. The end of the side are called the polygon vertices. Triangle is the most simple form of polygon having three side and three vertices. The polygon may be of any shape.
- 2. Types of Polygon 2 1.Convex 2. Concave 3. Complex
- 3. Convex 3 Convex polygon isa polygon in which if we take any points which are surely inside the polygon and if we draw a joining these two points and if all the points on that line lies into the polygon, then such a polygon is called as convex polygon.
- 4. Concave 4 Concave polygon isa polygon in which if we take any two points which are surely inside the polygon and if we draw a line joining these two points and if all the points on that line are not lying inside the polygon, then such a polygon is called as concave polygon.
- 5. Complex 5 In a computergraphics, a polygon which is neither convex nor concave is referred as complex polygons. The complex polygon includes any polygon which intersects itself of the polygon which overlaps itself. The complex polygon has a boundary.
- 6. Polygon Representation 6 OP XY 6 5 5 2 7 5 2 9 3 2 7 1 2 5 1 2 3 3 2 5 5 A(5,5) F(3,3) E(5,1) D(7,1) B(7,5) C(9,3)
- 7. Inside Test Methods 7 1.Even-odd Method 2. Winding Number Method
- 8. Even - OddMethods 8 x4,y4 x3,y3 x2,y2 x1,y1 Count even Outside Point ‘B’ A Fig. 3.2.2 Fig. 3.2.3 Fig. 3.2.4 Outside Point ‘B’ AC Count Odd Fig. 3.2.5 ACB A B C D A
- 9. Winding Number Methods 9 1 -11 Fig. 3.2.7 Fig. 3.2.8 Fig. 3.2.9 AB C 1 Newly drawn line +1 -1 DC B A
- 10. Winding Number Methods 10 Fig.3.2.10 (c) Fig. 3.2.10 (a) Fig. 3.2.10 (b) -1 -1+1 D AB C Case I Case II Case III B C A D -1 -1+1 B C D A -1-1 +1
- 11. Polygon Filling 11 Polygon Filling Algorithm 1.Boundary Fill Algorithm 2. Flood Fill (Seed Fill) Algorithm 3. Edge Fill Algorithm 4. Fence Fill Algorithm 1. Scan Line Algorithm Pixel Level Geometric Level
- 12. 4-connected Method 12 In this4 neighboring points of a current test point are tested. These are pixel positions that are right, left, above and below of current pixels as shown in figure 2.26 8-connected Method Here 8 neighboring pixels of current test pixel are tested. These pixel positions are left, right, above and 4 – diagonal positions of current pixel as shown in figure 2.27
- 13. Boundary Fill Algorithm 13 bfill(x, y, newcolor) { current = getpixel (x, y); if (current != newcolor)) && (current != boundarycolor) { putpixel (x, y, newcolor); bfill (x+1, y, newcolor); bfill (x-1, y, newcolor); bfill (x, y+1, newcolor); bfill (x, y-1, newcolor); } } The following procedure illustrate a recursive method for boundary fill by using 4-connected method.
- 14. Flood Fill (Seedfill) Algorithm 14 F-fill (x, y, newcolor) { current = getpixel (x, y); if (current != newcolor) { putpixel (x, y, newcolor); f-fill (x-1, y, newcolor); f-fill (x+1, y, newcolor); f-fill (x, y-1, newcolor); f-fill (x, y+1, newcolor); } } The following procedure illustrate a recursive method for flood fill by using 4-connected method.
- 15. Scan line Algorithm 15 (0,0) ymin y max A(x1, y1) B(x2,y2) C(x3, y3) D(x4, y4) Fig. 3.4.1 Edge y max x max y min x min Slope AB y2 x2 y1 x1 m1 AD y4 x4 y1 x1 m2 CD y3 x4 Y4 x4 m3 BC y2 x2 y3 x3 m4 Fig. 3.4.1
- 16. Scan line Algorithm 16 Edgey max x max y min x min Slope AB y2 x2 y1 x1 m1 BC y2 x2 y3 x3 m4 CD y3 x3 Y4 x4 m3 AD y4 x4 y1 x1 m2 Fig. 3.4.2 y max =y2 y2-2 y2-2 y min A(x1,y1) B(x2, y2)
- 17. Scan line Algorithm 17 Fig.3.4.3 Fig. 3.4.4 Edge y max x max y min x min Slope AB y1 x1 y2 x2 m1 BC y1 x1 y5 x3 m2 CD y3 x3 Y2 x2 m3 AD y x y x m4 A(x1,y1) D(x4,y4) C(x3,y3) B(x2,y2) max min A (x1,y1) C (x3,y3) D(x4,y4) B (x2,y2) E (x5,y5) Y max Y min
- 18. Scan line Algorithm 18 Fig.3.4.5 Fig. 3.4.6 A C D B E P2 P3 P4P1 yp P3 P2P1
- 19. Flood fill VsBoundary fill Vs Edge fill Vs Fence fill Vs Scan line fill 19 Flood fill Boundary Fill Edge fill Fence Fill Scan line fill Need Seed point to start. Need Seed point to start. Based on complimenting the pixels. Based on complimenting the pixels. Based on line drawing, polygon is filled. Current pixels color is compared with new pixel color. Current pixels color is compared with new pixel color and boundary color. Pixels which are on right side of an edge are getting complemented. Pixels which are on right side of an edge and to the left of fence as well as left side of edge and right side of fence are getting complemented. Intersection of polygon edges are found with the scan line and then the solid lines are drawn between two such intersection points. Useful for polygons having single color boundary. Useful for polygons having multi color boundary. More number of pixels are unnecessarily accessed. Less number of pixels are accessed as compared to edge fill. Need separate attention to handle concave polygons.
- 20. Connected boundary fillAlgorithm boundary_fill (x, y, f_colour, b_colour) { if (getpixel (x, y) != b_colour && getpixel (x, y) != f_colour) { putpixel (x, y, f_colour); boundary_fill (x+1, y, f_colour, b_colour); boundary_fill (x-1, y, f_colour, b_colour); boundary_fill (x, y+1, f_colour, b_colour); boundary_fill (x, y-1, f_colour, b_colour); boundary_fill (x+1, y+1, f_colour, b_colour); boundary_fill (x-1, y-1, f_colour, b_colour); boundary_fill (x+1, y-1, f_colour, b_colour); boundary_fill (x-1, y+1, f_colour, b_colour); } } To enhance speed of filling colour one may look for alternative of 4-connected. In this algorithm all adjacent pixels will be consider for filling till the match is true. Algorithm is as follows : 20